Black lines in imported UV layout
UV layout is just a visual guide for texturing and there is no reason to keep it for the actual texture.

Basic UV layout workflow

The principles are the same for all editing tools on how to use the uv-layout no matter it is Inkscape, Photoshop, Affinity Designer, Gimp or whatever:


Export your UV-Layout out of Blender (UV Editor > UV > Export UV Layout), either as vector svg or image file (png)
Open up the file in your favourite editing tool
Create a new layer and give it any meaningful name like 'UV-Layout'
Place your UV-Layout onto that new layer and drag it on top of the stack
Set 'Blend-Mode' of the UV layer to 'Multiply' which basically allows to "see trough" the 'unfold geometry' and having the layout as reference in place:

To avoid moving the layer by accident lock your 'UV-Layout' layer by clicking the little lock-icon
Create your textures somewhere below that layer eg. in another layer called 'texture'
When done with texturing: disable your 'UV-Layout' layer by clicking the eye or stack icon left to the lock icon and export it as image file (eg. png, exr or jpg)
Back in Blender, load up the image in the Image Editor (Image > Open) to check for issues:


UV Editor vs Image Editor 
Final step is to assign the image texture to the shader and set the viewport shading to Material Preview or even to Rendered to see the actual result: